Difference between revisions of "Delta cosh"

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Let $p \in C_{rd}$ and $-\mu p^2$ be a [[regressive function]]. Then the $\Delta$ hyperbolic cosine function is defined by
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Let $p \in C_{rd}$ and $-\mu p^2$ be a [[regressive function]]. Then the $\Delta$-hyperbolic cosine function is defined by
 
$$\cosh_p(t,s) = \dfrac{e_p(t,s)+e_{-p}(t,s)}{2}.$$
 
$$\cosh_p(t,s) = \dfrac{e_p(t,s)+e_{-p}(t,s)}{2}.$$
=Properties=
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> Let $p\in C_{rd}$. If $-\mu p^2 \in \mathcal{R}$, then
 
$$\cosh^{\Delta}_p = p\sinh_p,$$
 
where $\sinh_p$ is the [[Delta sinh | $\Delta$-hyperbolic sine]] function.
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
=Relation to other functions=
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<div align="center">
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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<gallery>
<strong>Theorem:</strong> $\cosh^2_p - \sinh^2_p = e_{-\mu p^2}$
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File:Integercosh,a=0.6,s=0plot.png|Graph of $\cosh_{0.6}(t,0;\mathbb{Z})$.
<div class="mw-collapsible-content">
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</gallery>
<strong>Proof:</strong> █
 
</div>
 
 
</div>
 
</div>
  
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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=Properties=
<strong>Theorem:</strong> Let $\gamma$ be a nonzero regressive real number, then a general solution of the second order dynamic equation is
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[[Derivative of delta cosh]]<br />
$$y^{\Delta \Delta}-\gamma^2 y= 0$$
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[[Delta cosh minus delta sinh]]<br />
is given by
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[[Delta hyperbolic trigonometric second order dynamic equation]]<br />
$$y(t) = c_1 \cosh_{\gamma}(t,s) + c_2 \sinh_{\gamma}(t,s).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
 
=Examples=
 
=Examples=
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<center>{{:Delta special functions footer}}</center>
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[[Category:specialfunction]]
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[[Category:Definition]]

Latest revision as of 14:13, 28 January 2023

Let $p \in C_{rd}$ and $-\mu p^2$ be a regressive function. Then the $\Delta$-hyperbolic cosine function is defined by $$\cosh_p(t,s) = \dfrac{e_p(t,s)+e_{-p}(t,s)}{2}.$$

Properties

Derivative of delta cosh
Delta cosh minus delta sinh
Delta hyperbolic trigonometric second order dynamic equation

Examples

Time Scale $\Delta$-$\cosh_1$ Functions
$\mathbb{T}=$ $\cosh_1(t,0)=$
$\mathbb{R}$ $\cosh_1(t,0)=\cosh(t)$
$\mathbb{Z}$
$h\mathbb{Z}$ $\cosh_1(t,0)=\dfrac{1}{2}\left( (1-h)^{\frac{t}{h}} + (1+h)^{\frac{t}{h}}\right) = \displaystyle\sum_{k=0}^{\infty} h_{2k}(t,0) $
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$

$\Delta$-special functions on time scales


$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$